L(s) = 1 | − 2.55·2-s + 3-s + 4.51·4-s − 2.55·6-s + 4.19·7-s − 6.40·8-s + 9-s + 4.51·12-s + 4.93·13-s − 10.7·14-s + 7.32·16-s + 6.10·17-s − 2.55·18-s − 2.51·19-s + 4.19·21-s − 3.08·23-s − 6.40·24-s − 12.5·26-s + 27-s + 18.9·28-s − 3.60·29-s − 1.99·31-s − 5.87·32-s − 15.5·34-s + 4.51·36-s − 7.64·37-s + 6.40·38-s + ⋯ |
L(s) = 1 | − 1.80·2-s + 0.577·3-s + 2.25·4-s − 1.04·6-s + 1.58·7-s − 2.26·8-s + 0.333·9-s + 1.30·12-s + 1.36·13-s − 2.86·14-s + 1.83·16-s + 1.48·17-s − 0.601·18-s − 0.575·19-s + 0.915·21-s − 0.643·23-s − 1.30·24-s − 2.46·26-s + 0.192·27-s + 3.57·28-s − 0.668·29-s − 0.358·31-s − 1.03·32-s − 2.67·34-s + 0.751·36-s − 1.25·37-s + 1.03·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9075 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9075 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.579864388\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.579864388\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + 2.55T + 2T^{2} \) |
| 7 | \( 1 - 4.19T + 7T^{2} \) |
| 13 | \( 1 - 4.93T + 13T^{2} \) |
| 17 | \( 1 - 6.10T + 17T^{2} \) |
| 19 | \( 1 + 2.51T + 19T^{2} \) |
| 23 | \( 1 + 3.08T + 23T^{2} \) |
| 29 | \( 1 + 3.60T + 29T^{2} \) |
| 31 | \( 1 + 1.99T + 31T^{2} \) |
| 37 | \( 1 + 7.64T + 37T^{2} \) |
| 41 | \( 1 + 2.95T + 41T^{2} \) |
| 43 | \( 1 - 0.186T + 43T^{2} \) |
| 47 | \( 1 + 5.56T + 47T^{2} \) |
| 53 | \( 1 - 6.21T + 53T^{2} \) |
| 59 | \( 1 + 0.530T + 59T^{2} \) |
| 61 | \( 1 - 8.14T + 61T^{2} \) |
| 67 | \( 1 - 6.64T + 67T^{2} \) |
| 71 | \( 1 + 11.0T + 71T^{2} \) |
| 73 | \( 1 + 0.0446T + 73T^{2} \) |
| 79 | \( 1 - 12.5T + 79T^{2} \) |
| 83 | \( 1 - 6.46T + 83T^{2} \) |
| 89 | \( 1 + 0.535T + 89T^{2} \) |
| 97 | \( 1 - 11.0T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.912149816498623758535207925720, −7.50983072294175822727107759610, −6.71451692347305627815141687336, −5.87091141719892233051568858260, −5.13091822639424630229756360911, −3.97469758849330640298337187828, −3.25549281779618318317214526277, −2.01861099139052904532916212104, −1.67064103838720909001629619906, −0.826511767767227397374714600019,
0.826511767767227397374714600019, 1.67064103838720909001629619906, 2.01861099139052904532916212104, 3.25549281779618318317214526277, 3.97469758849330640298337187828, 5.13091822639424630229756360911, 5.87091141719892233051568858260, 6.71451692347305627815141687336, 7.50983072294175822727107759610, 7.912149816498623758535207925720